One elegant way to overcome this problem, is to reduce the dimension of the eigenspace, along the line of spectral nonlinear reduction techniques. Following the Laplacian eigenmaps scheme [1]}, one can perform a generalized eigendecomposition of the Laplacian matrix \({L}\)  and use the \(K\)  smallest eigenvaluesleaving out the eigenvalue 0 and its unitary eigenvector with their associated eigenspace (\(K\ll N\)). We denote by \({U}_x^{K}\)  the \(N\times K\) block matrix of \({U}_x\) . The columns of \({U}_x^{K}\)  correspond to the \(K\)  eigenvectors associated with the selected eigenvalues, while its rows represent the coordinates of the set \(\mathcal {\bar{X}}\)  in the embedded \(K\) -dimensional space: \(\mathcal {\bar{X}}=\lbrace x_1,\ldots ,x_N\rbrace \) . Since \(X_{i}\in \mathbb {R}^{3}\)  and \(x_{i}\in \mathbb {R}^{K}\) , this corresponds to a mapping \(\mathbb {R}^{3}\mapsto \mathbb {R}^{K}\) . Respectively, for the second shape \(\mathcal {Y}\) , we can use the matrix \({U}_y^{K}\)  to derive the embedding \(\mathcal {\bar{Y}}=\lbrace y_1,\ldots ,y_M\rbrace \) .
